Interference-free LMS-based adaptive asynchronous receiver

ABSTRACT

The invention relates to an interference-free LMS-based asynchronous receiver for digital transmission and recording systems. The receiver, having an asynchronously placed LMS-based adaptive equalizer, has 2 control loops: a timing recovery loop (by means of, for instance a PLL (Phase locked loop) and an equalizer&#39;s adaptation loop. Interference between the two loops is avoided by deriving a condition the equalizer should fulfill to avoid the interference between the two loops, which implies “orthogonal control functionality” and by combining the condition with the equalizer&#39;s adaptation loop. The equalizer shall adapt so that the condition is always true.

This application is a 371 of PCT/IB03/01605 Apr. 15, 2003.

FIELD OF THE INVENTION

The invention generally relates to digital transmission and recordingsystems. In particular, it relates to a receiver for delivering a datasequence a_(k) at a data rate 1/T from a received sequence r_(n) sampledat a clock rate 1/Ts, asynchronous to the data rate 1/T.

The invention also relates to a digital system comprising a transmitterfor transmitting a digital sequence via a channel and a receiver forextracting said digital sequence from said channel, wherein saidreceiver is a receiver as described above. The invention further relatesto an equalizer adaptation method for said receiver. It finally relatesto a computer program product for such a receiver and to a signal forcanying said computer program.

The invention applies to a wide variety of asynchronous receivers foruse in digital transmission and recording systems. It is particularlyadvantageous in high density/capacity optical disc systems such as theBlu-ray Disc system (BD).

BACKGROUND ART

U.S. Pat. No. 5,999,355 describes an asynchronous receiver such as theone mentioned in the opening paragraph. In accordance with the citedpatent, the equalizer is a tapped delay line (Finite Impulse Responsefilter) with a tap spacing of Ts seconds. Control of the equalizer isbased on the classical LMS (Least Mean Square) algorithm; that is tosay, correlating the tap sequences with a suitable engr sequenceproduces updates of the equalizer tap values. Classical LMS techniquesnormally apply to synchronous receivers wherein error and tap sequenceshave the same sampling rate and are phase synchronous. The asynchronousreceiver described in the cited patent thus comprises at Least twoprovisions in order that error and tap sequences have the same samplingrate and are phase synchronous. The latter condition implies that anylatency in the error sequence should be matched by delaying the tapsequences accordingly. The aforementioned two provisions include aninverse sampling rate conversion (ISRC) for converting the synchronousewor sequence at the data rate 1/T into an equivalent error sequence ofsampling rate 1/Ts. The receiver, having an asynchronously placedLMS-based adaptive equalizer, has two control loops: a

U.S. Pat. No. 5,999,355 describes an asynchronous receiver such as theone mentioned in the opening paragraph. In accordance with the citedpatent, the equalizer is a tapped delay line (Finite Impulse Responsefilter) with a tap spacing of Ts seconds. Control of the equalizer isbased on the classical LMS (Least Mean Square) algorithrti; that is tosay, correlating the tap sequences with a suitable error sequenceproduces updates of the equalizer tap values. Classical LMS techniquesnormally apply to synchronous receivers wherein error and tap sequenceshave the same sampling rate and are phase synchronous. The asynchronousreceiver described in the cited patent thus comprises at least twoprovisions in order that error and tap sequences have the same samplingrate and are phase synchronous. The latter condition implies that anylatency in the error sequence should be matched by delaying the tapsequences accordingly. The aforementioned two provisions include aninverse sampling rate conversion (ISRC) for converting the synchronouserror sequence at the data rate 1/T into an equivalent error sequence ofsampling rate 1/Ts. The receiver, having an asynchronously placedHISbased adaptive equalizer, has two control loops: a timing recoveryloop or PLL (Phase locked loop) and an equalizer's adaptation loop.Unless precautions are taken, the two loops can interfere with eachother, which may lead to instability.

SUMMARY OF THE INVENTION

It is an object of the invention to provide an asynchronous receiverusing an alternative adaptation topology that circumvents thedisadvantage mentioned above. In accordance with the invention, areceiver as mentioned in the opening paragraph is provided, comprising:

-   -   an adaptive equalizer (EQ) for delivering an equalized sequence        (y_(n)) from said received sequence (r_(n)), said equalizer        operating at the clock rate 1/Ts and being controlled via an        equalizer's adaptation loop,    -   a sampling rate converter (SRC1) for converting said equalized        sequence (y_(n)) into an equivalent input sequence (x_(k)) to be        provided to an error generator (21) at the data rate 1/T via a        timing recovery loop,    -   an error generator (21) for delivering, from said input sequence        (x_(k)), the data sequence (a_(k)) and an error sequence (e_(k))        to be used in both loops,    -   orthogonal control functionality means (40) for deriving a        condition for the adaptive equalizer (EQ) to fulfill in order to        decrease interference between said equalizer's adaptation loop        and said timing recovery loop.

In this way, the interference is avoided between the two loops byderiving a condition the equalizer should obey in order to delete theinterference between the two loops, which implies “orthogonal controlfunctionality” and by combining the condition with the equalizer'sadaptation loop. The equalizer shall adapt so that the condition isalways true. With the orthogonal control functionality means, theequalizer is adapted by the equalizer's adaptation controlling algorithm(based on Least Mean Square algorithm or LMS for instance) such that theorthogonal control functionality condition is obeyed, resulting in aninterference-free system.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention and additional features, which may be optionally used toimplement the invention, are apparent from and will be elucidated withreference to the drawings described hereinafter and wherein:

FIG. 1 is a functional block diagram illustrating a generic asynchronousreceiver topology for use in digital transmission and recording systems,

FIG. 2 is a functional block diagram illustrating an example of anasynchronous LMS-based receiver topology.

FIG. 3 is a functional block diagram illustrating a further example of areceiver in accordance with an asynchronous LMS-based topology,

FIG. 4 is a functional block diagram illustrating a receiver topology inaccordance with a first embodiment of the invention,

FIG. 5 a and FIG. 5 b are graphs illustrating simulation resultsrelating to the receivers of FIGS. 3 and 4, respectively,

FIG. 6 is a graph illustrating linear interpolation in accordance with asecond embodiment of the invention,

FIG. 7 is a functional block diagram illustrating a receiver topology inaccordance with a second embodiment of the invention,

FIG. 8 a and FIG. 8 b are graphs illustrating simulation resultsrelating to the receiver of FIG. 7,

FIG. 9 is a schematic block diagram illustrating a digital system inaccordance with the invention.

DETAILED DESCRIPTION OF THE DRAWINGS

The following remarks relate to reference signs. Identical block labelsin all Figures usually indicate the same functional entities. In thesequel, we will also adopt the convention that vectors are denoted byunderlined symbols, and that the symbols k and n refer to sequences ofsampling rates 1/T and 1/Ts, respectively. For example, according tothis convention the notation a_(k) refers to a scalar sequence ofsampling rate 1/T, and the notation S_(n) refers to a vector sequence ofsampling rate 1/Ts. The length of a vector will be denoted by the symbolN with a subscript indicating the symbol used for the vector.Accordingly, for example, the length of the vector S_(n) is denotedN_(s).

FIG. 1 illustrates a generic topology of an asynchronous base-bandreceiver for digital transmission and recording systems. The receivergenerates a data sequence a_(k) at a data rate 1/T from a receivedsignal r(t). The received signal r(t) is applied to an analog low passfilter LPF whose main function is to suppress out-of-band noise. Ananalog-to-digital converter ADC, which operates at a crystal-controlledfree-running sampling rate 1/Ts, asynchronous to the data rate 1/T,which is high enough to prevent aliasing, digitizes the LPF output. TheADC output is applied to an equalizer EQ which serves to conditioninter-symbol interference and noise. The equalizer operates at thesampling rate 1/Ts, i.e. asynchronously to the data rate 1/T. Asampling-rate converter SRC produces an equivalent synchronous outputwhich serves as the input of a bit detector DET for delivering the datasequence a_(k). The SRC forms part of a timing-recovery loop (TRL),which is not depicted explicitly in FIG. 1. Asynchronous and synchronousclock domains are indicated in FIG. 1 with the symbols 1/Ts and 1/T,respectively.

To cope with variations of the system parameters, the equalizer EQ oftenneeds to be adaptive. To this end, error information is extracted fromthe bit detector DET by an error formation circuit EFC and is used tocontrol (update) the equalizer taps via a control module CTL. This formsan equalizer's adaptation loop (EAL). Error formation occurs in thesynchronous (1/T) clock domain, while control necessarily occurs in theasynchronous (1/Ts) domain. In between, an inverse sampling-rateconverter ISRC is required. In practice, the equalizer is often a tappeddelay line (Finite Impulse Response filter) with a tap spacing of Tsseconds.

Existing asynchronous adaptation techniques are based on LMS (Least MeanSquare) algorithms. With LMS, cross-correlating the tap sequences with asuitable error sequence derives updated information for the equalizertaps. For this to work, the tap and error signals need to be synchronousboth in sampling rate and in phase. The first condition is met via theISRC. The second one requires that the total latency of SRC,bit-detector, error formation circuit, and ISRC is matched by delayingthe tap signals accordingly, prior to cross-correlation. Both ISRC anddelay matching add to the complexity of the solution. Delay matching,moreover, may not be accurate because of the time-varying nature of thelatency of SRC and ISRC. As a result, adaptation performance maydegrade.

FIG. 2 shows an example of a receiver comprising an adaptation topologythat overcomes the disadvantages mentioned before. Only a portion of thedata receiver is shown in FIG. 2, namely the portion relevant to digitalequalizer adaptation. In particular, the timing-recovery subsystem ofthe receiver, which controls the sampling-rate converter SRC and thetemporal interpolation means TI, is not shown. The receiver comprises anadaptive equalizer EQ, a pair of sampling-rate converters SRC1 and SRC2,and a detector DET to produce a data sequence a_(k) from a receivedinput sequence r_(n). The detector DET is part of an error generator 21which generates an error sequence e_(k) to be used in the equalizer'scontrol loop from the bit decisions generated by the bit detector.Adaptation of the equalizer is based on LMS techniques as described, forexample, in J. W. M. Bergmans: “Digital Baseband Transmission andRecording”, published by Kluwer Academic Publishers, Boston, 1996,denoted [ref.]. Central to these techniques is that tap updateinformation is produced by correlating the tap signals (receivedsequence r_(n)) with the error signals. Error and tap signals shouldhave the same sampling rate, and should moreover be phase synchronous;any latency in the error signal should be matched by delaying the tapsignals accordingly.

In FIG. 2, r_(n) denotes the sequence obtained by periodic sampling of,for example, an analog replay signal from a recording channel. Samplingis performed at a free-running clock rate 1/Ts which is generally notequal to the data rate 1/T. The sequence r_(n) is passed through anequalizer EQ having Ts-spaced taps w_(n) for producing an equalizedsequence y_(n) at its output. Preferably, the equalizer EQ is an FIR(Finite Impulse Response) transversal filter, but it may be anyequalizer that contains a linear combiner. The purpose of the equalizeris to shape the response of the (for example recording) channel to aprescribed target response and to condition the noise spectrum. Theequalizer EQ is followed by a sample rate converter SRC which transformsthe Ts-spaced equalized sequence y_(n) into an equivalent T-spacedsequence x_(k) to be provided at the input of an error generator 21comprising a bit detector DET. The T-spaced input sequence x_(k) isideally synchronous to the data rate 1/T of the channel data sequencea_(k). Actually, the bit detector DET produces estimates â_(k) of thechannel bits a_(k). Assuming that the bit detector produces correctdecisions, the data sequence a_(k) and its estimate â_(k) are identical.Therefore, the outputs of the bit detector are denoted a_(k) in allFigures. Occasional bit errors do not significantly affect theperformance of the system. Alternatively, at the beginning oftransmission, a predetermined data sequence (often referred to aspreamble) may precede the actual data in order for initial adaptation tobe based on a replica of this predetermined data sequence, which may besynthesized locally in the data receiver without any bit errors. It iscommon practice to perform the initial stage of adaptation in thisso-called ‘data-aided’ mode of operation, and to switch to the‘decision-directed’ mode of operation as depicted in FIG. 2 onceadaptation loops have converged. Though not depicted explicitly in FIG.2, it is to be understood that the present description also pertains tothis ‘data-aided’ mode of operation.

The remaining part of FIG. 2 illustrates the mechanism of the controlloop for adaptively updating the equalizer tap coefficient vectorsequence W_(n) using LMS techniques. All digital operations involved inthe control loop can be realized, for example, by a microprocessorcarrying out a suitable computer program. The double-line arrows betweenblocks indicate vector signals transfers while single arrows indicatescalar signals. Therefore, the control loop (equalizer's adaptation)comprises:

-   -   a second sampling rate converter SRC2 for converting a delayed        version of the received sequence r_(n) into an intermediate        control sequence I_(k) at the data rate 1/T, this second SRC,        denoted SRC2, being preferably the same as the first SRC1,    -   control information production means 22 for deriving a        synchronous control vector sequence Z_(k) at the data rate 1/T,        from the error sequence e_(k) and the intermediate control        sequence i_(k), and    -   temporal interpolation means TI for deriving the control vector        sequence S_(n) from said synchronous control vector sequence        Z_(k).

In FIG. 2, the control vector sequence S_(n) directly controls theequalizer, i.e. the equalizer tap vector sequence W_(n) simply coincideswith S_(n). The synchronous control vector sequence Z_(k) produced bythe control information production means is formed by a bank of Nzintegrators 22, whose input is derived from a cross product 24 e_(k).I_(k), where I_(k) is an intermediate vector sequence consisting of Niintermediate sequences. All the vector lengths are equal. Therefore,Nz=Ni=2M+1, the number 2M+1 being the number of taps w_(n) in theequalizer EQ. This intermediate vector sequence I_(k) is derived fromthe received sequence r_(n). A predefined delay τ is applied to thereceived sequence r_(n). The delayed version of the received sequencer_(n) is provided to a sampling rate converter SRC2 to form anintermediate sequence i_(k), prior to a shift register SR performing aserial to parallel conversion to form the intermediate vector sequenceI_(k) from the intermediate sequence i_(k).

The input of the equalizer is thus converted to the data rate domainafter it has been delayed with a predefined delay. The predefined delaydoes not vary with time and is well known. It is equal to the amount ofdelay from the input of the equalizer to the output. Once both signals,i.e. the signals at the output of each sampling rate converter, are inthe data rate domain, the equalizer coefficient updates can be easilycomputed. The adaptation scheme will be detailed below.

The variables at the output of the integrators 22, denoted z_(k) ^(j),obey the following equation:z _(k+1) ^(j) =z _(k) ^(j)+μΔ_(k) ^(j) , j: −M, . . . ,M  (1)where:

-   -   z_(k) ^(j) is the output of the j-th integrator at instant k,    -   μ is a small scaling factor (often referred to as step size)        which determines closed-loop time constants,    -   Δ_(k) ^(j) is a tap-error estimate at iteration k, and    -   2M+1 is the number of taps of the equalizer.

According to the LMS scheme, the estimate Δ_(k) ^(j) is given by:Δ_(k) ^(j) =e _(k) ·i _(k-j) , j: −M, . . . ,M  (2)where:

-   -   e_(k) is the error between the SRC output and a (delayed version        of) the desired detector input d_(k)=(a*g)_(k), with:    -   g_(k) the target response (of a filter G) for the equalizer        adaptation    -   i_(k-j) is a delayed version of the received sequence r_(n)        converted into the data rate 1/T.

For the sake of completeness it is mentioned that equation (2) and FIG.2 describe only one of the various possible manners to derive tap-errorestimates Δ_(k) ^(j) from the error sequence e_(k) and the inputsequence r_(n). For example, both of the two sequences e_(k) and r_(n)may be strongly quantized so as to simplify implementation, and themultiplication in (2) may be replaced by a selective-update mechanism.

FIG. 2 shows that the synchronous control vector sequence Z_(k) at theoutput of the integrators is updated every T seconds (synchronousdomain), while the equalizer coefficient vector W_(n) needs to beupdated every Ts seconds, since the equalizer operates in theasynchronous domain. The necessary time-base conversion is performedthrough the temporal interpolation means TI for deriving an asynchronouscontrol vector sequence S_(n) at the sampling rate 1/Ts from thesynchronous control vector sequence Z_(k) at the output of the bank ofintegrators. Since tap values change only slowly with respect to bothsampling rates, the temporal interpolation can be done in the simplestconceivable manner, for example, via a bank of latches performingzeroth-order interpolation. When T_(s) deviates too much from T, anadditional issue is raised, which requires an additional functionality,called spatial interpolation. The additional functionality is describedwith reference to FIG. 3.

The equalizer has a tap spacing of Ts seconds, i.e. it acts to delay theinput sequence in steps of Ts seconds to obtain the successive tapsignals, which are then combined linearly with weights w_(n) ^(j), j:−M, . . . ,M, that are defined by the coefficient vector sequence W_(n).The control vector sequence s_(n) at the output of the bank ofintegrators, however, pertains to a T-spaced equalizer, i.e. successivecomponents s^(j), j: −M, . . . ,M, of s_(n) are meant in principle asweighting factors for an equalizer with tap spacing T. The discrepancybetween this nominal tap spacing of T seconds and the actual tap spacingof Ts seconds results in a degradation of adaptation performance, bothin terms of the steady-state solution at which the equalizer settles andin terms of a degradation of loop efficiency. As a result, the topologyof FIG. 2 is mainly suitable for near-synchronous applications, forexample, applications in which 1/Ts and 1/T are close to each other, andpreferably differ by less than approximately 20 s⁻. This condition ismet in many practical systems, for example, in most channel ICs(Integrated Circuits) for hard disk drivers (optical storage).

In order to be able to use the invention within a larger range ofapplications, an improvement of the scheme described in FIG. 2 isproposed in FIG. 3. According to this improvement, the control loopfurther comprises spatial conversion means for deriving the equalizercoefficient vector sequence W_(n) from the asynchronous control vectorsequence S_(n) at the output of the temporal interpolation means. As aresult, an initially T-spaced sequence generated within the control loopis converted into an equivalent Ts-spaced sequence for controlling theequalizer coefficient vector W_(n). In FIG. 3, these spatial conversionmeans are indicated with the symbol SI. Since the update variables s_(n)^(j) describe the coefficients of a T-spaced equalizer, it is indeednecessary to convert this T-spaced information into Ts-spacedinformation. This necessitates interpolation on the coefficients s^(j),which is performed by the Spatial Interpolator block SI. Conceptually,the update variables s^(j) are T-spaced samples of an underlyingtime-continuous equalizer filter whose impulse response is denoted w(t),i.e. s^(j)=w^(i)=w(jT), j: −M, . . . ,M. Assuming that w(t) wereavailable, we would have to resample it at positions t_(i)=i×Ts, for i:−M, . . . ,M, in order to generate the necessary equalizer coefficientsw^(i)=w(i×Ts). The variable t here does not indicate time but position,and assumes continuous values from a certain interval (the span of thefilter). In the same sense, i is a position index that is independent oftime, i.e. t_(i) is fully determined by i and does not change over time.However, since only T-spaced samples of w(t), namely s^(j), areavailable, interpolation of these samples must be used to produce theTs-spaced variables w^(i).

One of the simplest forms of interpolation is linear interpolation,which is attractive from a computational point of view, but other formsof interpolation may be considered such as, for example,nearest-neighbor interpolation, which is even simpler. The re-samplingpositions t_(i)=i×T_(s) can be equivalently written ast_(i)=(m_(i)+c_(i))T, where 0≦c_(i)<1, and

$\begin{matrix}{{m_{i} = \left\lbrack {i\;\frac{T_{s}}{T}} \right\rbrack},{c_{i} = {{i\;\frac{T_{s}}{T}} - {m_{i}.}}}} & (3)\end{matrix}$

As c_(i) varies between 0 and 1, t_(i) varies between m_(i)T and(m_(i)+1)T, and w(t) varies between w(m_(i)T)=s^(m) _(i) andw((m_(i)+1)T)=s^(m) ^(i) ⁺¹ _(i). According to one method of linearinterpolation, the value of w(t) at position t_(i) is then calculatedas:w _(i) =w(t _(i))=(1−c _(i))×s ^(m) ^(i) +c _(i) ×s ^(m) ^(i) ⁺¹  (4)

With the aid of (4), the spatial interpolator SI of FIG. 3 converts theT-spaced taps s^(i) at the output of the latch to Ts-spaced tap settingsw^(i) representing the equalizer taps. In order to perform thisconversion it is necessary to know, or estimate, the ratio Ts/T of thechannel bit rate to the sampling rate as indicated in equation (3).However, an estimate of this ratio is already available within thesampling rate converter SRC1 of FIG. 3. The SRC re-samples the Ts-spacedsequence y_(n) at instants t_(k)=kT, which can be re-written ast_(k)=(m_(k)+μ_(k))Ts.

In the presence of phase errors, the difference between successivesampling instants varies from the nominal value of T according tot_(k)−t_(k−1)=T+τ_(k)T, where τ_(k) is a phase error in thereconstructed T-spaced clock. Then we arrive at the following equation:

$\begin{matrix}{{\left( {m_{k} - m_{k - 1}} \right) + \left( {\mu_{k} - \mu_{k - 1}} \right)} = {\frac{T}{T_{s}} + {\tau_{k}\frac{T}{T_{s}}}}} & (5)\end{matrix}$

The timing-recovery loop that controls the SRC1 acts to force theaverage of the phase error to zero. Therefore, the average of thequantity on the left-hand side of (5) will settle on the actual value ofT/Ts, or the inverse of the ratio that is needed for linearinterpolation.

For the adaptive equalizer of FIG. 3 solutions are needed in order torealize “orthogonal” control functionality of the timing recovery loopor PLL and the equalizer's adaptation loop. Time-shifts in the impulseresponse of the FIR filter (Finite Impulse Response filter) of theequalizer (EQ), which occur as offsets in the “group delay” are fullycompensated by the timing recovery loop. The “group delay” indicates thederivative of the phase characteristic of the filter. As a result theerror e_(k) is independent of the offset in the group delay, which maylead to divergence. In order to avoid interference between the timingrecovery and the equalizer's adaptation loop, the impulse response ofthe adaptive equalizer must not contain a linear phase term in frequencyv, which is called an orthogonal functionality condition. In fact, thePLL should be solely responsible for the correction of linear phase termdistortions and the adaptive equalizer for all higher-order phasedistortions, for example, in optical storage systems, like v² for adefocus term, v³ for a tilt or coma term, v⁴ for a residual sphericalaberration term due to low-frequency variations in the cover layerthickness, for example.

The novel LMS-based asynchronous equalizer with orthogonal controlfunctionality extension adapts the equalizer in accordance with theequalizer's control algorithm, for example, of the LMS type, providedthat the condition defined above is fulfilled in every adaptation step.A receiver in accordance with a first embodiment of the invention isillustrated in FIG. 4. In the topology of FIG. 4, we assume that Ts isalmost equal to T. Same functional entities as in FIGS. 2 and 3 areindicated by same letter references. The new receiver comprisesorthogonal control functionality means to derive a condition theequalizer should obey in order to decrease interference between the twoloops. The equalizer shall adapt so that the condition is always true.

The derivation orthogonal functionality condition is explained below.Let us denote the transfer function of the FIR filter w_(k) by W(v):

$\begin{matrix}{{W(v)} = {\sum\limits_{k}{w_{k}\exp\left\{ {2{\pi\mathbb{i}}\;{vk}} \right\}\mspace{14mu}\left( {{- 0.5} \leq v \leq 0.5} \right)}}} \\{= {{A(v)}\exp\left\{ {{\mathbb{i}\varphi}(v)} \right\}}}\end{matrix}$

The phase φ(v) must not contain terms linear in v, the group delayoffset must be constrained:

${\frac{\mathbb{d}{\varphi(v)}}{\mathbb{d}v}❘_{v = 0}} = 0$

Further, we know that, since the equalizer coefficients w_(k) are real,we have for A(v):A(−v)=A(v)

And thus also that

${\frac{\mathbb{d}{A(v)}}{\mathbb{d}v}❘_{v = 0}} = 0$ Since$\frac{\mathbb{d}{W(v)}}{\mathbb{d}v} = {{\frac{\mathbb{d}{A(v)}}{\mathbb{d}v}\exp\left\{ {{\mathbb{i}\varphi}(v)} \right\}} + {i\frac{\mathbb{d}{\varphi(v)}}{\mathbb{d}v}{W(v)}}}$

Combining the equations results in:

${\frac{\mathbb{d}{W(v)}}{\mathbb{d}v}❘_{v = 0}} = 0$

Which is equivalent to:

${\sum\limits_{k}{kw}_{k}} = 0$

A filter that leaves linear phase terms untouched should have taps w_(k)obeying this constraint. To prevent the interference between theequalizer adaptation loop and the tiring recovery, the objective of thefilter adaptation algorithm must be altered to minimize the mean squareerror power J, where E[x] indicates the expectation value of thestatistical variable x:J(k)=E[e _(k) ²](Basic Least Mean Square (LMS) algorithm) subject to the additionalcondition:

${\sum\limits_{k}{kw}_{k}} = 0$

This results in a new cost function (the criterion, which is to beminimized by the adaptation is called the cost function), where {tildeover (J)}(k) means:

${\overset{\sim}{J}(k)} = {{J(k)} + {\lambda \cdot \left( {\sum\limits_{k}{kw}_{k}} \right)}}$where λ is a Lagrange multiplier. This multiplier must be chosen suchthat {tilde over (J)}(k) is minimized as a function of the filter tapsw_(p):

${{\overset{\sim}{\nabla}}_{p}(k)} = {\frac{\partial{\overset{\sim}{J}(k)}}{\partial w_{p}} = {0\mspace{14mu}{\forall p}}}$or${{\overset{\sim}{\nabla}}_{p}(k)} = {{{\lambda \cdot p} + {2 \cdot {E\left\lbrack {\left( {{\sum\limits_{j}{x_{k - j}w_{j}}} - {\sum\limits_{j}{g_{k - j}{\hat{a}}_{j}}}} \right) \cdot x_{k - p}} \right\rbrack}}} = {0\mspace{14mu}{\forall p}}}$

To determine λ, the energy in

$\sum\limits_{p}{{\overset{\sim}{\nabla}}_{p}(k)^{2}}$must be minimized:

${\frac{\partial}{\partial\lambda}\left( {\sum\limits_{p}{{\overset{\sim}{\nabla}}_{p}(k)^{2}}} \right)} = {{\sum\limits_{p}{p \cdot {{\overset{\sim}{\nabla}}_{p}(k)}}} = 0}$Resulting  in:$\lambda = {\frac{{- 2} \cdot {E\left\lbrack {\left( {{\sum\limits_{j}{x_{k - j}w_{j}}} - {\sum\limits_{j}{g_{k - j}{\hat{a}}_{j}}}} \right) \cdot {\sum\limits_{a}{nx}_{k - a}}} \right\rbrack}}{\sum\limits_{n}n^{2}} = \frac{{- 2} \cdot {E\left\lbrack {e_{k} \cdot {\sum\limits_{n}{nx}_{k - n}}} \right\rbrack}}{\sum\limits_{n}n^{2}}}$where â_(k) stands for the decisions on a_(k).

The equalizer adaptation loop must now solve the equation:

${\underset{\_}{\overset{\sim}{\nabla}}(k)} = {\frac{\partial{\overset{\sim}{J}(k)}}{\partial{\overset{\_}{w}(k)}} = {\left( {\frac{\partial{\overset{\sim}{J}(k)}}{\partial{w_{\kappa}(k)}},\ldots\mspace{11mu},\frac{\partial{\overset{\sim}{J}(k)}}{\partial{w_{\kappa}(k)}}} \right)->0}}$

Iteratively, by using the steepest-descent method:

${w_{p}\left( {k + 1} \right)} = {{w_{p}(k)} - {\mu{{\overset{\sim}{\nabla}}_{p}(k)}}}$resulting  in: $\begin{matrix}{{{\overset{\sim}{\nabla}}_{p}(k)} = {\frac{\partial{\overset{\sim}{J}(k)}}{\partial{w_{p}(k)}} = {{E\left\lbrack {2 \cdot e_{k} \cdot \frac{\partial e_{k}}{\partial{w_{p}(k)}}} \right\rbrack} + \frac{{- 2} \cdot p \cdot {E\left\lbrack {e_{k} \cdot {\sum\limits_{n}{nx}_{k - n}}} \right\rbrack}}{\sum\limits_{n}n^{2}}}}} \\{= {{2 \cdot {E\left\lbrack {e_{k} \cdot x_{k - p}} \right\rbrack}} + \frac{{- 2} \cdot p \cdot {E\left\lbrack {e_{k} \cdot {\sum\limits_{n}{nx}_{k - n}}} \right\rbrack}}{\sum\limits_{n}n^{2}}}}\end{matrix}$From the practical point of view the steepest descent update mentionedabove is not computable: the expectation instruction requires an averagecomputation over a very long period. Therefore, the gradient is replacedby an instantaneous gradient, which gives:

${w_{p}\left( {k + 1} \right)} = {{w_{p}(k)} - {2 \cdot \mu \cdot e_{k} \cdot x_{k - p}} + {2 \cdot \mu \cdot p \cdot e_{k} \cdot \frac{\sum\limits_{n}{nx}_{k - n}}{\sum\limits_{n}n^{2}}}}$

This altered LMS algorithm will minimize the average mean square errorpower. Instead of the power, one may seek to minimize the averageabsolute value of the error:

$\begin{matrix}{{{\overset{\sim}{\nabla}}_{p}(k)} = {\frac{\partial{\overset{\sim}{J_{ll}}(k)}}{\partial{w_{p}(k)}} = {{E\left\lbrack {2 \cdot {{sign}\left( e_{k} \right)} \cdot \frac{\partial e_{k}}{\partial{w_{p}(k)}}} \right\rbrack} + \frac{{- 2} \cdot p \cdot {E\left\lbrack {{{sign}\left( e_{k} \right)} \cdot {\sum\limits_{n}{nx}_{k - n}}} \right\rbrack}}{\sum\limits_{n}n^{2}}}}} \\{= {{2 \cdot {E\left\lbrack {{{sign}\left( e_{k} \right)} \cdot x_{k - p}} \right\rbrack}} + \frac{{- 2} \cdot p \cdot {E\left\lbrack {{{sign}\left( e_{k} \right)} \cdot {\sum\limits_{n}{nx}_{k - n}}} \right\rbrack}}{\sum\limits_{n}n^{2}}}}\end{matrix}$leading to a new sign-algorithm:

${w_{p}\left( {k + 1} \right)} = {{w_{p}(k)} - {2 \cdot \mu \cdot {{sign}\left( e_{k} \right)} \cdot x_{k - p}} + {2 \cdot \mu \cdot p \cdot {{sign}\left( e_{k} \right)} \cdot \frac{\sum\limits_{n}{nx}_{k - n}}{\sum\limits_{n}n^{2}}}}$

The multiplication by sign(e_(k)) involves only sign reversals and isthus significantly simpler. As a sanity check, the condition for theorthogonal control functionality of the timing recovery and theequalizer adaptation loop shall be evaluated:

${\sum\limits_{k}{kw}_{k}} = 0$for the algorithm:

${w_{p}\left( {k + 1} \right)} = {{w_{p}(k)} - {2 \cdot \mu \cdot e_{k} \cdot x_{k - p}} + {2 \cdot \mu \cdot p \cdot e_{k} \cdot \frac{\sum\limits_{n}{nx}_{k - a}}{\sum\limits_{n}n^{2}}}}$

The increment does obey the condition:

${\sum\limits_{p}{p \cdot \left( {{{- 2} \cdot \mu \cdot e_{k} \cdot x_{k - p}} + {2 \cdot \mu \cdot p \cdot e_{k} \cdot \frac{\sum\limits_{n}{nx}_{k - n}}{\sum\limits_{n}n^{2}}}} \right)}} = {{2 \cdot \mu \cdot e_{k} \cdot \left( {{\sum\limits_{p}{px}_{k - p}} - {\sum\limits_{p}{p^{2} \cdot \frac{\sum\limits_{n}{nx}_{k - n}}{\sum\limits_{n}n^{2}}}}} \right)} = 0}$The same check can be done for the sign-algorithm, leading to a similarresult.

FIG. 5 illustrates simulation results showing the gradient of theequalizer coefficients versus time. FIG. 5 a shows results obtained withthe topology of FIG. 3 and FIG. 5 b with the topology of FIG. 4,including the orthogonal control functionality. A comparison between thetwo Figures shows the importance of the orthogonal control functionalityimplemented in FIG. 4. In these simulations, a 5-tap adaptive equalizerhas been used, and the Ts domain is 2% deviated with respect to the datarate domain. FIGS. 5 a and 5 b illustrate the gradients of the 5 filtertaps. In FIG. 5 a the compensation for orthogonal functionality isabsent, causing the taps to diverge. On the other hand in FIG. 5 b thetopology of FIG. 4 was employed, causing the equalizer taps to converge.

Now another issue is addressed in the topology of FIG. 6 relating to asecond embodiment of the invention, which applies even when Ts iscompletely different from T. The filter updates are produced in the datarate domain and are meant for a T-spaced equalizer. However, this filteris Ts-spaced. This topology is thus primarily useful fornear-synchronous applications, where Ts differs not too much from T.

Conceptually the calculated T-spaced filter updates need to be convertedinto the Ts-domain, necessitating an interpolation. FIG. 3 and itsrelated description give details on the spatial interpolator, denotedSI. This interpolator performs a linear interpolation, which is veryattractive from a computational point of view.

The re-sampling positions t_(i)=iT_(s) can be written ast_(i)=(m_(i)+c_(i))T, where 0≦c_(i)<1 and

$m_{i} = {{{{floor}\left( {i\;\frac{T_{s}}{T}} \right)}\mspace{14mu} c_{i}} = {{i\;\frac{T_{s}}{T}} - m_{i}}}$

As c_(i) varies between 0 and 1, t_(i) varies between m_(i)T and(m_(i)+1)T, and w varies between w(m_(i)T) and w((m_(i)+1)T).

According to the linear interpolation as shown in FIG. 6, we get:W(iT _(s))=(1−c _(i))·w(m _(i) T)+c _(i) ·w((m _(i)+1)T)

Starting from the basic LMS algorithm:w _(m) _(i) (k+1)=w _(m) _(i) (k)−2·μ·sign(e _(k))·x _(k-m) _(i)

Applying the spatial interpolator:{tilde over (w)} _(i)(k+1)=(1−c _(i))·w _(m) _(i) (k+1)+c _(i) ·w _(m)_(i) ₊₁(k+1)

Leads to:{tilde over (w)} _(i)(k+1)=(1−c _(i))·(w _(m) _(i) (k)−2·μ·sign(e_(k))·x _(k-m) _(i) )+c _(i)·(w _(m) _(i) ₊₁(k)−2·μ·sign(e _(k))·x_(k-m) _(i) ⁻¹)={tilde over (w)} _(i)(k)−2·μ·sign(e _(k))·└(1−c _(i))·x_(k-m) _(i) ⁻¹ ┘={tilde over (w)} ₁(k)−2·μ·sign(e _(k))·{tilde over (x)}_(k,1)

Time shifts in the impulse response of this new FIR are fullycompensated by the timing recovery. As a result the error e_(k) isindependent of the offset in the group delay, which may lead todivergence. In order to have orthogonal control functionality of thetiming recovery and the equalizer's adaptation loop, one has to includethe results derived in the previous description in relation to FIG. 4.

This results in the topology of FIG. 7.

${{\overset{\sim}{w}}_{i}\left( {k + 1} \right)} = {{{\overset{\sim}{w}}_{1}(k)} - {2 \cdot \mu \cdot {{sign}\left( e_{k} \right)} \cdot \left\lbrack {{\left( {1 - c_{i}} \right) \cdot x_{k - m_{i}}} + {c_{i} \cdot x_{k - m_{i} - 1}}} \right\rbrack} + \mspace{130mu}{2 \cdot \mu \cdot {{sign}\left( e_{k} \right)} \cdot i \cdot \frac{\sum\limits_{n}{n\left\lbrack {{\left( {1 - c_{n}} \right) \cdot x_{k - m_{n}}} + {c_{n} \cdot x_{k - m_{n} - 1}}} \right\rbrack}}{\sum\limits_{n}n^{2}}}}$

If the spatial interpolator were to be applied to the ‘upgraded’ versionof LMS algorithm, which has the extra term for orthogonal functionality,the equalizer's adaptation loop would still have stability problems. Theasynchronous FIR filter would not obey the condition:

${\sum\limits_{k}{k\;{\overset{\sim}{w}}_{k}}} = 0.$

This would be due to the fact that the spatial interpolator does notpreserve this property when there are at least 5 equalizer taps. One canintuit this easily, for example, for a 5-tap filter one has:

$\begin{matrix}\underset{︸}{\begin{matrix}{{\overset{\sim}{w}}_{- 2} = \frac{w_{- 2} + w_{- 1}}{2}} & \; \\{{\overset{\sim}{w}}_{- 1} = {\frac{w_{0}}{4} + \frac{3 \cdot w_{- 1}}{4}}} & \; \\{{\overset{\sim}{w}}_{0} = w_{0}} & {{{and}\mspace{14mu}{\sum\limits_{k}{k\; w_{k}}}} = 0} \\{{\overset{\sim}{w}}_{1} = {\frac{w_{0}}{4} + \frac{3 \cdot w_{1}}{4}}} & \; \\{{\overset{\sim}{w}}_{2} = \frac{w_{2} + w_{1}}{2}} & \;\end{matrix}} \\{{\sum\limits_{k}{k\;{\overset{\sim}{w}}_{k}}} = {{{- w_{- 2}} - \frac{5 \cdot w_{- 1}}{4} + \frac{5 \cdot w_{1}}{4} + w_{2}} \neq 0}}\end{matrix}$

In order to have an idea of the necessity of the spatial interpolator,DVR simulations are done with ideal signals. In the DVR optical receiverthe relation T/Ts amounts to 4/3, leading to an adaptive equalizer with¾ T spacing.

FIG. 8 illustrates simulation results showing the evolution of the FIRcoefficients versus time. FIG. 8 a shows results obtained with thetopology of FIG. 7. In FIG. 8 b the spatial interpolator is absent: theT spaced tap updates are connected to the ¾T-spaced equalizer withoutany conversion. This causes the taps to diverge. In FIG. 8 a, on theother hand, the topology of FIG. 7 was employed, which causes theequalizer to converge.

FIG. 9 shows an example of a system in accordance with the inventioncomprising a receiver as shown in FIGS. 2, 3, 4 and 7. The system maybe, for example, a digital recording system. It comprises a recorder 41for recording a digital sequence 93 on a recording support 92 and areceiver 94 for reading the recorded sequence 95 from said recordingsupport. The recording support 92 may be, for example, an optical disk.

An interference-free least mean square based asynchronous equalizationtopology has been described for preventing interference between thetiming recovery loop and the equalizer's adaptation loop. Sincetime-shifts in the impulse response of the FIR filter, which occur asoffsets in the group delay, are fully compensated by the timingrecovery, the error e_(k), which drives the filter's adaptation loop,would be independent on the offset in the groupdelay, which may lead todivergence.

It has been described how the interference between the adaptation loopsin an asynchronous LMS based equalizer can be avoided. First a conditionis derived which provides the asynchronous adaptive equalizer withorthogonal control functionality. Subsequently this constraint isincorporated in the least mean square criterion employing a Lagrangemultiplier. This leads to a topology that has cancelled out theinterference between the adaptation loops. Finally, this new structureis extended with a spatial interpolator in order to have moreflexibility regarding the T/Ts ratio.

Although the invention was described above by way of example withreference to a particular LMS-based asynchronous receiver topologyillustrated in FIG. 4 and FIG. 7, this does not limit the scope of theinvention. The basic principle of the invention, called the “orthogonalcontrol functionality”, is also applicable to any LMS-based asynchronousreceiver topology having a timing recovery loop and an adaptiveequalizer's adaptation loop. The basic principle of the invention is thesolution for avoiding interference between the timing recovery loop andthe equalizer's adaptation loop, where the equalizer is placed in theasynchronous domain. This solution, described with reference to FIG. 4and FIG. 7, is in this case calculated for the LMS-based control but mayin fact also be calculated for a zero forcing control. The solutionconsists of altering the adaptation algorithm such that the conditionfor interference-free working of the system is fulfilled.

The drawings and their description hereinbefore illustrate rather thanlimit the invention. It will be evident that there are numerousalternatives which fall within the scope of the appended claims. In thisrespect, the following closing remarks are made. There are numerous waysof implementing functions by means of items of hardware or software, orboth. In this respect, the drawings are very diagrammatic, eachrepresenting only one possible embodiment of the invention. Thus,although a drawing shows different functions as different blocks, thisby no means excludes either that a single item of hardware or softwarecarries out several functions, or that a function can be carried out byan assembly of items of hardware or software, or both.

1. A receiver for delivering a data sequence (a_(k)) at a data rate 1/Tfrom a received sequence (r_(n)) sampled at a clock rate 1/Ts,asynchronous to the data rate 1/T, the receiver comprising: an adaptiveequalizer (EQ) for delivering an equalized sequence (y_(n)) from saidreceived sequence (r_(n)), said equalizer operating at the clock rate1/Ts and being controlled via an equalizer's adaptation loop, a samplingrate converter (SRC1) for converting said equalized sequence (y_(n))into an equivalent input sequence (x_(k)) to be provided to an errorgenerator (21) at the data rate 1/T via a timing recovery loop, an errorgenerator (21) for delivering, from said input sequence (x_(k)), thedata sequence (a_(k)) and an error sequence (e_(k)) to be used in bothloops, orthogonal control functionality means (40) for deriving acondition for the adaptive equalizer (EQ) to fulfill in order todecrease interference between said equalizer's adaptation loop and saidtiming recovery loop.
 2. A receiver as claimed in claim 1, wherein thecontrol loop further comprises spatial conversion means (SI) forconverting a given initially T-spaced sequence generated within thecontrol loop into an equivalent Ts-spaced sequence for controlling saidequalizer coefficient vector (W_(n)).
 3. A receiver as claimed in claim2, wherein said spatial conversion means (SI) are arranged to perform alinear interpolation.
 4. A receiver as claimed in claim 2, wherein saidspatial conversion means (SI) are arranged to perform a nearest-neighborinterpolation.
 5. A digital system comprising a transmitter fortransmitting a digital sequence via a channel support and a receiver forextracting said digital sequence from said channel support, wherein saidreceiver is a receiver as claimed in claim
 1. 6. In a receivercomprising an adaptive equalizer, an equalizer adaptation method ofreceiving a sequence (r_(n)), sampled at a clock rate 1/Ts, and ofdelivering a data sequence (a_(k)) at a data rate 1/T, the methodcomprising the following steps: an adaptive equalizing step ofdelivering an equalized sequence (y_(n)) from the received sequence(r_(n)) using an equalizer coefficient vector (W_(n)) in a control loop,a first sampling rate converting step (SRC1) of converting saidequalized sequence (y_(n)) into an equivalent input sequence (x_(k)) tobe processed through an error generating step (21) at the data rate 1/Twithin a timing recovery loop, an error generating step (21) ofgenerating, from said input sequence (x_(k)), the data sequence (a_(k))and an error sequence (e_(k)) at the data rate 1/T to be used in bothloops, a step of generating a control vector sequence (S_(n)) from theerror sequence (e_(k)) and the received sequence (r_(n)), forcontrolling said equalizer coefficient vector (W_(n)), an orthogonalcontrol step (40) for deriving a condition for the adaptive equalizer tofulfill in order to decrease interference between said control loop andthe timing recovery loop.
 7. A computer program product for a receivercomputing a set of instructions which when loaded into the receiver,causes the receiver to carry out the method as claimed in claim
 6. 8. Asignal for carrying a computer program, the computer program beingarranged to carry out the method as claimed in claim 6.